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Theorem lsmelvalm 15285
Description: Subgroup sum membership analog of lsmelval 15283 using vector subtraction. TODO: any way to shorten proof? (Contributed by NM, 16-Mar-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
Hypotheses
Ref Expression
lsmelvalm.m  |-  .-  =  ( -g `  G )
lsmelvalm.p  |-  .(+)  =  (
LSSum `  G )
lsmelvalm.t  |-  ( ph  ->  T  e.  (SubGrp `  G ) )
lsmelvalm.u  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
Assertion
Ref Expression
lsmelvalm  |-  ( ph  ->  ( X  e.  ( T  .(+)  U )  <->  E. y  e.  T  E. z  e.  U  X  =  ( y  .-  z ) ) )
Distinct variable groups:    y, z,  .-    y, G, z    ph, y,
z    y, T, z    y, U, z    y, X, z
Allowed substitution hints:    .(+) ( y, z)

Proof of Theorem lsmelvalm
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 lsmelvalm.t . . 3  |-  ( ph  ->  T  e.  (SubGrp `  G ) )
2 lsmelvalm.u . . 3  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
3 eqid 2436 . . . 4  |-  ( +g  `  G )  =  ( +g  `  G )
4 lsmelvalm.p . . . 4  |-  .(+)  =  (
LSSum `  G )
53, 4lsmelval 15283 . . 3  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  ->  ( X  e.  ( T  .(+)  U )  <->  E. y  e.  T  E. x  e.  U  X  =  ( y
( +g  `  G ) x ) ) )
61, 2, 5syl2anc 643 . 2  |-  ( ph  ->  ( X  e.  ( T  .(+)  U )  <->  E. y  e.  T  E. x  e.  U  X  =  ( y ( +g  `  G ) x ) ) )
72adantr 452 . . . . . . . 8  |-  ( (
ph  /\  y  e.  T )  ->  U  e.  (SubGrp `  G )
)
8 eqid 2436 . . . . . . . . 9  |-  ( inv g `  G )  =  ( inv g `  G )
98subginvcl 14953 . . . . . . . 8  |-  ( ( U  e.  (SubGrp `  G )  /\  x  e.  U )  ->  (
( inv g `  G ) `  x
)  e.  U )
107, 9sylan 458 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  T )  /\  x  e.  U )  ->  (
( inv g `  G ) `  x
)  e.  U )
11 eqid 2436 . . . . . . . . 9  |-  ( Base `  G )  =  (
Base `  G )
12 lsmelvalm.m . . . . . . . . 9  |-  .-  =  ( -g `  G )
13 subgrcl 14949 . . . . . . . . . . 11  |-  ( T  e.  (SubGrp `  G
)  ->  G  e.  Grp )
141, 13syl 16 . . . . . . . . . 10  |-  ( ph  ->  G  e.  Grp )
1514ad2antrr 707 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  T )  /\  x  e.  U )  ->  G  e.  Grp )
1611subgss 14945 . . . . . . . . . . . 12  |-  ( T  e.  (SubGrp `  G
)  ->  T  C_  ( Base `  G ) )
171, 16syl 16 . . . . . . . . . . 11  |-  ( ph  ->  T  C_  ( Base `  G ) )
1817sselda 3348 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  T )  ->  y  e.  ( Base `  G
) )
1918adantr 452 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  T )  /\  x  e.  U )  ->  y  e.  ( Base `  G
) )
2011subgss 14945 . . . . . . . . . . 11  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  ( Base `  G ) )
217, 20syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  T )  ->  U  C_  ( Base `  G
) )
2221sselda 3348 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  T )  /\  x  e.  U )  ->  x  e.  ( Base `  G
) )
2311, 3, 12, 8, 15, 19, 22grpsubinv 14864 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  T )  /\  x  e.  U )  ->  (
y  .-  ( ( inv g `  G ) `
 x ) )  =  ( y ( +g  `  G ) x ) )
2423eqcomd 2441 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  T )  /\  x  e.  U )  ->  (
y ( +g  `  G
) x )  =  ( y  .-  (
( inv g `  G ) `  x
) ) )
25 oveq2 6089 . . . . . . . . 9  |-  ( z  =  ( ( inv g `  G ) `
 x )  -> 
( y  .-  z
)  =  ( y 
.-  ( ( inv g `  G ) `
 x ) ) )
2625eqeq2d 2447 . . . . . . . 8  |-  ( z  =  ( ( inv g `  G ) `
 x )  -> 
( ( y ( +g  `  G ) x )  =  ( y  .-  z )  <-> 
( y ( +g  `  G ) x )  =  ( y  .-  ( ( inv g `  G ) `  x
) ) ) )
2726rspcev 3052 . . . . . . 7  |-  ( ( ( ( inv g `  G ) `  x
)  e.  U  /\  ( y ( +g  `  G ) x )  =  ( y  .-  ( ( inv g `  G ) `  x
) ) )  ->  E. z  e.  U  ( y ( +g  `  G ) x )  =  ( y  .-  z ) )
2810, 24, 27syl2anc 643 . . . . . 6  |-  ( ( ( ph  /\  y  e.  T )  /\  x  e.  U )  ->  E. z  e.  U  ( y
( +g  `  G ) x )  =  ( y  .-  z ) )
29 eqeq1 2442 . . . . . . 7  |-  ( X  =  ( y ( +g  `  G ) x )  ->  ( X  =  ( y  .-  z )  <->  ( y
( +g  `  G ) x )  =  ( y  .-  z ) ) )
3029rexbidv 2726 . . . . . 6  |-  ( X  =  ( y ( +g  `  G ) x )  ->  ( E. z  e.  U  X  =  ( y  .-  z )  <->  E. z  e.  U  ( y
( +g  `  G ) x )  =  ( y  .-  z ) ) )
3128, 30syl5ibrcom 214 . . . . 5  |-  ( ( ( ph  /\  y  e.  T )  /\  x  e.  U )  ->  ( X  =  ( y
( +g  `  G ) x )  ->  E. z  e.  U  X  =  ( y  .-  z
) ) )
3231rexlimdva 2830 . . . 4  |-  ( (
ph  /\  y  e.  T )  ->  ( E. x  e.  U  X  =  ( y
( +g  `  G ) x )  ->  E. z  e.  U  X  =  ( y  .-  z
) ) )
338subginvcl 14953 . . . . . . . 8  |-  ( ( U  e.  (SubGrp `  G )  /\  z  e.  U )  ->  (
( inv g `  G ) `  z
)  e.  U )
347, 33sylan 458 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  T )  /\  z  e.  U )  ->  (
( inv g `  G ) `  z
)  e.  U )
3518adantr 452 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  T )  /\  z  e.  U )  ->  y  e.  ( Base `  G
) )
3621sselda 3348 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  T )  /\  z  e.  U )  ->  z  e.  ( Base `  G
) )
3711, 3, 8, 12grpsubval 14848 . . . . . . . 8  |-  ( ( y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) )  ->  (
y  .-  z )  =  ( y ( +g  `  G ) ( ( inv g `  G ) `  z
) ) )
3835, 36, 37syl2anc 643 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  T )  /\  z  e.  U )  ->  (
y  .-  z )  =  ( y ( +g  `  G ) ( ( inv g `  G ) `  z
) ) )
39 oveq2 6089 . . . . . . . . 9  |-  ( x  =  ( ( inv g `  G ) `
 z )  -> 
( y ( +g  `  G ) x )  =  ( y ( +g  `  G ) ( ( inv g `  G ) `  z
) ) )
4039eqeq2d 2447 . . . . . . . 8  |-  ( x  =  ( ( inv g `  G ) `
 z )  -> 
( ( y  .-  z )  =  ( y ( +g  `  G
) x )  <->  ( y  .-  z )  =  ( y ( +g  `  G
) ( ( inv g `  G ) `
 z ) ) ) )
4140rspcev 3052 . . . . . . 7  |-  ( ( ( ( inv g `  G ) `  z
)  e.  U  /\  ( y  .-  z
)  =  ( y ( +g  `  G
) ( ( inv g `  G ) `
 z ) ) )  ->  E. x  e.  U  ( y  .-  z )  =  ( y ( +g  `  G
) x ) )
4234, 38, 41syl2anc 643 . . . . . 6  |-  ( ( ( ph  /\  y  e.  T )  /\  z  e.  U )  ->  E. x  e.  U  ( y  .-  z )  =  ( y ( +g  `  G
) x ) )
43 eqeq1 2442 . . . . . . 7  |-  ( X  =  ( y  .-  z )  ->  ( X  =  ( y
( +g  `  G ) x )  <->  ( y  .-  z )  =  ( y ( +g  `  G
) x ) ) )
4443rexbidv 2726 . . . . . 6  |-  ( X  =  ( y  .-  z )  ->  ( E. x  e.  U  X  =  ( y
( +g  `  G ) x )  <->  E. x  e.  U  ( y  .-  z )  =  ( y ( +g  `  G
) x ) ) )
4542, 44syl5ibrcom 214 . . . . 5  |-  ( ( ( ph  /\  y  e.  T )  /\  z  e.  U )  ->  ( X  =  ( y  .-  z )  ->  E. x  e.  U  X  =  ( y ( +g  `  G ) x ) ) )
4645rexlimdva 2830 . . . 4  |-  ( (
ph  /\  y  e.  T )  ->  ( E. z  e.  U  X  =  ( y  .-  z )  ->  E. x  e.  U  X  =  ( y ( +g  `  G ) x ) ) )
4732, 46impbid 184 . . 3  |-  ( (
ph  /\  y  e.  T )  ->  ( E. x  e.  U  X  =  ( y
( +g  `  G ) x )  <->  E. z  e.  U  X  =  ( y  .-  z
) ) )
4847rexbidva 2722 . 2  |-  ( ph  ->  ( E. y  e.  T  E. x  e.  U  X  =  ( y ( +g  `  G
) x )  <->  E. y  e.  T  E. z  e.  U  X  =  ( y  .-  z
) ) )
496, 48bitrd 245 1  |-  ( ph  ->  ( X  e.  ( T  .(+)  U )  <->  E. y  e.  T  E. z  e.  U  X  =  ( y  .-  z ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2706    C_ wss 3320   ` cfv 5454  (class class class)co 6081   Basecbs 13469   +g cplusg 13529   Grpcgrp 14685   inv gcminusg 14686   -gcsg 14688  SubGrpcsubg 14938   LSSumclsm 15268
This theorem is referenced by:  lsmelvalmi  15286  pgpfac1lem2  15633  pgpfac1lem3  15635  pgpfac1lem4  15636  mapdpglem3  32473
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-0g 13727  df-mnd 14690  df-grp 14812  df-minusg 14813  df-sbg 14814  df-subg 14941  df-lsm 15270
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